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Section 3-6. Counting. FUNDAMENTAL COUNTING RULE. For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m · n ways. This generalizes to more than two events. EXAMPLES.
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Section 3-6 Counting
FUNDAMENTAL COUNTING RULE For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m· n ways. This generalizes to more than two events.
EXAMPLES • How many two letter “words” can be formed if the first letter is one of the vowels a, e, i, o, u and the second letter is a consonant? • OVER FIFTY TYPES OF PIZZA! says the sign as you drive up. Inside you discover only the choices “onions, peppers, mushrooms, sausage, anchovies, and meatballs. There are also 3 different types of crust and 4 types of cheese. Did the advertisement lie? • Janet has five different books that she wishes to arrange on her desk. How many different arrangements are possible? • Suppose Janet only wants to arrange three of her five books on her desk. How many ways can she do that?
FACTORIALS NOTE: 0! is defined to be 1. That is, 0! = 1
FACTORIAL RULE A collection of n objects can be arranged in order n! different ways.
PERMUTATIONS A permutation is an ordered arrangement. A permutation is sometimes called a sequence.
PERMUTATION RULE(WHEN ITEMS ARE ALL DIFFERENT) The number of permutations (or sequences) of r items selected from n available items (without replacement) is denoted by nPr and is given by the formula
PERMUATION RULE CONDITIONS • We must have a total of ndifferent items available. (This rule does not apply if some items are identical to others.) • We must select r of the n items without replacement. • We must consider rearrangements of the same items to be different sequences. (The arrangement ABC is the different from the arrangement CBA.)
EXAMPLE Suppose 8 people enter an event in a swim meet. Assuming there are no ties, how many ways could the gold, silver, and bronze prizes be awarded?
PERMUTATION RULE(WHEN SOME ITEMS ARE IDENTICAL TO OTHERS) If there are n items with n1 alike, n2 alike, . . . , nk alike, the number of permutations of all n items is
EXAMPLE How many different ways can you rearrange the letters of the word “level”?
COMBINATIONS A combination is a selection of objects without regard to order.
COMBINATIONS RULE The number of combinations of r items selected from n different items is denoted by nCr and is given by the formula NOTE: Sometimes nCr is denoted by
COMBINATIONS RULE CONDITIONS • We must have a total of n different items available. • We must select r of those items without replacement. • We must consider rearrangements of the same items to be the same. (The combination ABC is the same as the combination CBA.)
EXAMPLES • From a group of 30 employees, 3 are to be selected to be on a special committee. In how many different ways can the employees be selected? • If you play the New York regional lottery where six winning numbers are drawn from 1, 2, 3, . . . , 31, what is the probability that you are a winner? • Exercise 34 on page 183, [15-18] page 188.
EXAMPLE Suppose you are dealt two cards from a well-shuffled deck. What is the probability of being dealt an “ace” and a “heart”?
PERMUTATIONS VERSUS COMBINATIONS When different orderings of the same items are to be counted separately, we have a permutation problem, but when different orderings are not to be counted separately, we have a combination problem.